Lay the puff pastry sheet out onto a lightly floured work surface. Roll out slightly to smooth out any creases from the folds. Invert a 12-inch skillet over the top of the puff pastry. If the skillet is wider than the puff pastry, roll it out a bit more so that is it wider than the skillet. Using a knife, trim around the outside edge of the skillet. Place the cut out puff pastry round on a sheet tray lined with parchment paper. Prick with a fork. Cover with plastic wrap and refrigerate until ready to use.
Preheat the oven to 400 degrees F. Peel, core and slice each apple in eighths. If not using right away, place the apple pieces in a bowl of water with the juice of 1 lemon to keep from browning.
Place the 12-inch skillet used to measure the puff pastry over medium-high heat. Add the sugar, maple syrup, lemon juice and salt. Let cook until the sugar melts into a syrup and turns a light brown color. Turn the heat down to medium-low and carefully whisk in the butter. The mixture will vigorously bubble so be very careful. Whisk in the cinnamon and vanilla. Carefully arrange the apple slices cut-side up in the pan in a circular pattern. Let cook until the apples start to become slightly tender, about 10 minutes. Turn off the heat and place the puff pastry over the top of the apples, making sure to tuck the pastry inside the pan.
Bake in the oven until puff pastry is lightly browned and puffy, 20 to 25 minutes. Remove from the oven, let stand for 10 minutes, then invert onto a serving platter. The tart can also be made 8 hours in advance and be cooled completely. Just make sure to heat the pan over medium-low heat for 1 to 2 minutes to loosen the filling before inverting onto a platter.
Slice into eighths and serve with ice cream or whipped cream.